WorkSheet 2. Review of the Differential Equations Material

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1 WorkSheet 2 Review of the Differential Equations Material 1. Methods of solving differential equations. Separation of Variables Integrating factor Know solution to dy/ = ay and dy/ = ay, y(0) = c Know solution to dy/ = ay + b and dy/ = ay + b, y(0) = c Know Maple syntax for solving differential equations. 2. Exponential Growth/Decay problems Set up for continuous compounding problems (i.e. start from A(t+ t) A(t) and end with the differential equation da/ = ra). Know how to use the exponential equation to solve continuous compounding, bacteria growth, and radio active decay problem 3. Newton s Law of cooling Know the statement leading to the differential equation. Know how to use separation of variables (or simply know the solution since it is of the form / = ax + b) to solve dt/ = a(t (t) T s ) 4. Salt tank problems (one tank) Rate in = rate out (Easy to solve) Rate in 6= rate out (Can not use separation of variables - will need the integrating factor technique) 5. Falling Body problems. Assume that gravity is the only force (easy integration problem) Assume that wind resistance is also a force on the object (equation is now a differential equation). Know how to set up these problems in different coordinate systems 6. Interest rate problems Continuous compounding Depositing or withdrawing continuously 7. Direction Fields Simple ones by hand Using Maple Equilibrium solutions Stability of equilibrium solutions 8. Population problems Use the direction field to analyze Solve with Maple

2 9. Numerical Solutions Know how to derive Euler s method Know how to use Euler s method to generate a couple values of the function. Understand the maple document which gives other numerical methods Problems related to First Order DEs 1) What is the general solution to dy =3y? 2) What is the solution to = 2x, x(0) = 15? 3) What is the solution to = αx, x(0) = 5, x(2) = 1? 4) Use the separation of variable technique to find the general solution to dy =3y. 5) Use the separation of variable technique to find the general solution to dy =3x y. 6) Use the separation of variable technique to find the solution to =3t2 x 2,x(1) = 1. 7) A certain type of bacteria is known to grow at a rate proportional to the amount that is present. If you start with 2 grams of the bacteria and two hours later there are 2.1 grams, how much will you have after 5 hours? 8) Suppose that 15 grams of Einsteinium were released (the half life of Einsteinium is 270 days). How long will it take for the 15 grams to decay to 10 grams? 9) If glass of cold water (40 degrees) is placed in a room whose temperature is 70 degrees, how long will it take for the glass of water to heat up to 55 degrees? Use Newton s Law of Cooling and assume that the temperature after 10 minutes is 44 degrees. 10) Suppose that the situation in (9) is taking place on some other planet where the Law of Cooling state that the rate of change of the temperature is proportional to the square of the difference between the temperature of the object and the temperature of the surrounding medium. What DE describes the temperature on this new planet? 11) A tank 100 gallon tank contains a solution of water and salt with the initial amount of salt being 25 lbs. If pure water enters the tank at a rate of 5 gallons per minute and the mixtureleavesthetankatthesamerate,finheamountofsaltinthetankatanyfuture time t. 12)Acertaintypeofbacteriaisknowntogrowatarateproportionaltothesquarerootof theamountthatispresent. Ifyoustartwith3gramsofthebacteriaanhreehourslater there are 3.5 grams, how much will you have after 7 hours? 13) Suppose that 10 grams of radium were released (the half life of Radium is 1620 years). Howlongwillittakeforthe10gramstodecayto5grams?to2grams?

3 14) If a cup of hot chocolate (180 degrees) is placed in a room whose temperature is 70 degrees, how long will it take for the cup to cool to 100 degrees? Use Newton s Law of Cooling and assume that the temperature after 5 minutes is 150 degrees. 15) A 500 gallon tank contains a solution of water and salt with the initial amount of salt being 100 lbs. If pure water enters the tank at a rate of 10 gallons per minute and the mixtureleavesthetankatthesamerate,howlongwillittaketoflush the tank (assume that this means that 98% of the original salt is gone). What would the answer be if we started with S lbs of salt? 16) Suppose that you are on top of a 60 m building and you throw a ball upwards (from the edge so that it returns to the ground) with a speed of 50 m/sec. Now we will assume that gravity is the only force acting on the ball and also that the mass of the ball is 2 kg. First usenewton slaw(ma = P forces) to set up the problem as a differential equation. Then solve the problem (find the position x as a function t) and determine how high the ball goes and how long it is in the air. a)useapositioncoordinatesystemwhereupispositiveandwithoriginatthetopofthe building. b) Use a position coordinate system where up is positive and with origin on the ground. c) Use a position coordinate system where down is positive and with origin at the top of the building. 17) Suppose that you are on top of a 100 ft building and you throw a ball downwards (from the edge so that it goes to the ground) with a speed of 70 ft/sec. Suppose the ball weighs 2 lbs. First use Newton s Law ( ma = P forces) to set up the problem as a differential equation. Then solve the problem (find the position x as a function t) and determine how long it takes the ball to reach the ground. What is the ball s velocity when it hits the ground? 18) Suppose that you are on top of a 20 m building and you toss a 2 kg ball upwards (from the edge so that it returns to the ground) with a speed of 25 m/sec. Us a position coordinate system where up is positive and with origin on the ground. a) Assuming that only gravity acts on the ball, find how long the ball is in the air b) In addition to gravity, assume that there is a resistance force whose magnitude is 1.2 times the speed of the ball. Now determine how long the ball is in the air. 19) (a) A 500 gallon tank contains a solution of water and salt with the initial amount of salt being 100 lbs. If pure water enters the tank at a rate of 8 gallons per minute and the mixture leaves the tank at the same rate, how much salt is in the tank at time t? b) Same setup as (a) except there is salt in the water that enters the tank. Assuming that there is 1/2 lb of salt per gallon entering the tank, how much salt is in the tank at time t? 20) A 250 gallon tank contains a solution of water and salt with the initial amount of salt being 60 lbs. If pure water enters the tank at a rate of 5 gallons per minute and the mixture leaves the tank at the same rate, how much salt is in the tank at time t? 21) A 250 gallon tank contains a solution of water and salt with the initial amount of salt being 60 lbs. If pure water enters the tank at a rate of 5 gallons per minute and the mixture leaves the tank at the rate of 7 gallons per minute, how much salt is in the tank at time t?

4 22) A 250 gallon tank contains a solution of water and salt with the initial amount of salt being 60 lbs. If a brine solution containing 1/2 lb per gallon enters the tank at a rate of 5 gallons per minute and the mixture leaves the tank at the same rate, how much salt is in the tank at time t? 23) A 250 gallon tank contains a solution of water and salt with the initial amount of salt being 60 lbs. If a brine solution containing 1/2 lb per gallon enters the tank at a rate of 5 gallons per minute and the mixture leaves the tank at the rater of 7 gallons per minute, how much salt is in the tank at time t? 24) Solve dy =2y +4,y(0) = 3 in three different ways: separation of variables, integrating factor method, and using the structure of this non-homogeneous problem. 25) Even the simple exponential growth DEs can be solved using the integrating factor method. Solve =3x, x(0) = 8 using the integrating factor method. 26) Solve dy =2xy 3x 27) Solve dy = y + x2,y(0) = 2 28) Solve =2 x + t t 29) Solve = 2x +sin(t) 30) Solve dy +3y x =4 31) A 200 gallon tank initially contains 100 gallons of a solution of water and salt with the initial amount of salt being 25 lbs. If pure water enters the tank at a rate of 6 gallons per minute and the mixture leaves the tank at the rate of 4 gallons per minute, how much salt is in the tank at the time the tank overflows? Use whatever method is appropriate for solving this problem. Also give a graph of the solution. 32) A 200 gallon tank initially contains 100 gallons of a solution of water and salt with the initial amount of salt being 25 lbs. If a brine solution containing 1/4 lb per gallon enters thetankatarateof7gallonsperminuteanhemixtureleavesthetankattherater of 4 gallons per minute, how much salt is in the tank at the time the tank overflows? Use whatever method is appropriate for solving this problem. Also give a graph of the solution. 33) Suppose that you are on top of a 30 m building and you toss a 1.5 kg ball upwards (from the edge so that it returns to the ground) with a speed of 20 m/sec. Use a position coordinate system where up is positive and with the origin at the middle of the building. a) Assuming that only gravity acts on the ball, find how long the ball is in the air. b) In addition to gravity, assume that there is a resistance force whose magnitude is 0.8 times the speed of the ball. Now determine how long the ball is in the air. c) Change the resistive force in (b) so that the resistance force is 0.8 times the square of the speed of the ball. Now determine how long the ball is in the air. Be careful. I think this problem requires a careful setup.

5 34) (a) A 200 gallon tank contains a solution of water and salt with the initial amount of salt being 50 lbs. If pure water enters the tank at a rate of 4 gallons per minute and the mixture leaves the tank at 6 gallons per minute, how much salt is in the tank at time t? Sketch the solution. b) Same setup as (a) except there is salt in the water that enters the tank. Assuming that there is 1/2 lb of salt per gallon entering the tank, how much salt is in the tank at time t? Sketch the solution. 35) (a) Suppose that $1000 is put into a savings account at 8% interest compounded continuously. How much money is in the account at the end of 1 year, 1.5 years, and 5 years? (b) In addition to putting $1000 into a savings account at 8% interest compounded continuously, you decide to put another $1000 into the account throughout the year. Assume that this money is being put into the account continuously (for instance, you might be putting in $1000/52 per week this would almost be continuous). Now how much money is in the account at the end of 1 year, 1.5 years, and 5 years? 36) The radioactive isotope plutonium 241 decays so as to satisfy the differential equation = x where x is measured in milligrams and t in years. Determine the half life of plutonium ) Suppose that a person who is just about to retire has saved A 0 dollars while working the past few years. In order to invest this money at a 6% annual interest rate (compounded continuously) and be able to withdraw $20, 000 per year (assume that withdrawals are made continuously) for the next 20 years, what is the minimum value that A 0 can be? 38) A 300 gallon tank initially contains 200 gallons of a solution of water and salt with the initial amount of salt being 50 lbs. If a brine solution containing 1/2 lb per gallon enters the tank at a rate of 6 gallons per minute and the mixture leaves the tank at the rate of 4 gallons per minute, how much salt is in the tank at the time the tank overflows? Use whatever method is appropriate for solving this problem. Also give a graph of the solution. 39)GiventheIVP/ = f(t, x), x(0) = x 0, show how Euler s method gives an approximation x 1 to x(h) and x 2 = x(2h). Be sure to use a picture and clearly show the algorithm to find x 1 and x 2. 40) Find x 1 and x 2 using Euler s method given that / =2tx e x cos(t), x(0) = 1 using h = ) Find x 1 and x 2 using Euler s method given that / = t 2 +x 2,x(0) = 1 using h = ) An animal sanctuary had an initial population of 50 animals. After 2 years the population was 62, while after 4 years it was 76. Using the logistic population model, determine the equilibrium solution and the number of animals in the sanctuary after 20 years.